Problem 30 Find the square of each sum or d... [FREE SOLUTION]

Chapter 5: Problem 30

Find the square of each sum or difference. When possible, write down only the answer. $$(3-b)^{2}$$

Short Answer

Expert verified

The expanded form is \[ 9 - 6b + b^2 \].

Step by step solution

Identify the formula

Recognize that $ (3 - b)^2 $ represents a squared binomial, which can be expanded using the formula $ (a - b)^2 = a^2 - 2ab + b^2 $.

Identify the values of a and b

In the expression $ (3 - b)^2 $, the values are $ a = 3 $ and $ b = b $.

Apply the formula

Substitute $ a $ and $ b $ into the formula: $ 3^2 - 2(3)(b) + b^2 $.

Calculate the squares and products

Calculate the individual terms: $ 3^2 = 9, 2 \times 3 \times b = 6b $. This results in \[ 9 - 6b + b^2 \].

Write down the final expression

The expanded form of $ (3 - b)^2 $ is \[ 9 - 6b + b^2 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square of a Binomial

A binomial is an algebraic expression with two terms. Squaring a binomial means multiplying the binomial by itself. For example, the square of the binomial (3 - b) is written as (3 - b)^2. There is a formula to simplify this process, which is: (a - b)^2 = a^2 - 2ab + b^2. This formula helps us expand the binomial easily without needing to multiply the entire expression manually.

Algebraic Expansion

Algebraic expansion involves expressing a mathematical expression in an extended form. When we expand the binomial (3 - b)^2 , we use the formula a^2 - 2ab + b^2 to break it into simpler parts. We recognize a = 3 and b = b. Using the formula, we substitute these values into: a^2 - 2ab + b^2. This gives us 3^2 - 2(3)(b) + b^2. Calculating each term, we have 9 - 6b + b^2 . Thus, the expanded form is 9 - 6b + b^2.

Quadratic Expression

A quadratic expression is a polynomial with a degree of 2, meaning it includes a variable raised to the power of 2. In our expanded binomial (3 - b)^2, the result is 9 - 6b + b^2. Notice that 9 - 6b + b^2 is a quadratic expression with b^2 as the term with the highest degree. Quadratic expressions often appear in equations or functions that can be solved or simplified using algebraic methods. The typical form of a quadratic expression is ax^2 + bx + c. In our case, it's given by 9 - 6b + b^2, showcasing that the highest degree term determines the type of expression.

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91影视

Find the square of each sum or difference. When possible, write down only the answer. $$(3-b)^{2}$$

Short Answer

Step by step solution

Identify the formula

Identify the values of a and b

Apply the formula

Calculate the squares and products

Write down the final expression

Key Concepts

Square of a Binomial

Algebraic Expansion

Quadratic Expression

One App. One Place for Learning.

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